Shaking Things Up:
Statistical Modelling of Earthquakes

RSS Avon Local Group Meeting

Zak Varty

Talk Structure

  1. Seismology 101 & Earthquake Data
  2. Building Blocks of Statistical Seismology
  3. Current Work
  4. Future Directions

Aim: Know where we’re at, where we’re going, how to join in.

1: Seismology 101 & Earthquake Data

What is an Earthquake?

01:00

What is an Earthquake?

Vibrations in the ground:

  • Jumping up and down
  • Jack-hammer
  • Traffic and Trains
  • Sea Storms


Image by wirestock on Freepik

Natural Seismicity


  • Tectonic motion
  • Volcanic activity
  • Landslides


Volcanic Eruptions in Iceland. Image source - ABC.

Induced Seismicity

  • Pop concerts
  • Nuclear tests
  • Mine blasts
  • Fluid injection / extraction
  • Fracking


Lumen Field, Seattle. Image source - NME

Physics Recap

Cats on a Roof

What is a Fault?


Image Source - Wikimedia

Earthquakes as Fault Slip

Fault Slip

How does it happen?

  • Energy stored in rock under tension.
  • Movement when driving force exceeds static friction.
  • Depends on a lot of factors:
    • Material constants
    • Slip area and distance
    • Fault orientation

** What do we care about?**

  • Total energy released
  • Power (W = J/s)
  • Aseismic creep

What provides the driving force?

How do we measure earthquakes?

How do we locate earthquakes? (1)

How do we locate earthquakes? (2)

How do we locate earthquakes? (3)

How do we locate earthquakes? (4)

It’s a bit more complicated than that

  • 3 Dimensions

  • Energy dissipation

  • Background noise

  • Material Inhomogeneity

  • Reflection, refraction and interference

  • Expert interpretation

  • DL: Yoon et al (2023)

Earthquake Catalogues

  • x Easting / Longitude
  • y Northing / Latitude
  • z Depth (often nominal)
  • t Time (YYYY-MM-DD HH:MM:SS)
  • m Magnitude

A note on Magnitudes

Magnitudes are measured on logarithmic scales and typically reported to 1 decimal place.


\[\begin{align*} \text{Richter Scale} \rightarrow \text{Local Magnitude} &\sim \text{Amplitude}. \\ &\text{Moment Magnitude} %\sim \text{Energy}. \end{align*}\]


\[ E \propto A^{1.5} \text{ so } 10\times \text{ or } 32\times \text{ increase.}\]

What is an Aftershock?

01:00

What is an Aftershock?

2: Earthquake Modelling

Locations: Point Processes

  • Stochastic process \(\mathcal{X} = \{X_1, \ldots, X_N\}\)
    • Values represent locations in time / space, number of values N is also random.
  • Homogeneous Poisson Process on \(A\):
    • \(N(A) \sim \text{Pois}(\lambda |A|)\)
    • \((X_i | N(A) = n) \overset{\text{i.i.d}}{\sim} \text{Unif}(A).\)


Image source - Wikimedia

Locations: Inhomogeneous Poisson Processes

Introduce an intensity function \(\lambda(a): A \rightarrow \mathbb{R}^+_0\).

\[ N(A) \sim \text{Pois}\left(\Lambda(A)\right) \quad \text{where} \quad \Lambda(A) = \int_A \lambda(a) \mathrm{d}a.\]

Then

\[ X_i \overset{\text{i.i.d}}{\sim} f_X(a) = \frac{\lambda(a)}{\Lambda(a)}.\]

Physically motivated forms for \(\lambda(a)\), e.g. Bourne and Oates (2018): \[\Lambda \propto \exp(b_0 + b_1 z(a))\].

Locations: Adding in Aftershocks

Hawkes Processes add self excitation.

\[\begin{equation*} \lambda(t; \mathcal{H}_t) = \mu + \sum_{i: t_i < t} \alpha \exp\{-\beta(t - t_i)\}. \end{equation*}\]

Locations: Adding in Aftershocks - a better way

\[\begin{align*} \lambda(t) &= \mu + \sum_{i: t_i < t} \alpha \exp\{-\beta(t - t_i)\} \\ &= \mu + \sum_{i: t_i < t} \frac{\alpha}{\beta} \beta \exp\{-\beta(t - t_i)\} \\ &= \mu + \sum_{i: t_i < t} \alpha^\prime \beta \exp\{-\beta(t - t_i)\} \\ &= \mu + \sum_{i: t_i < t} \kappa(\cdot; \alpha^\prime) h(t - t_i; \beta)\}. \end{align*}\]

What about magnitudes?

Gutenberg-Richter Law:

\[ \log_{10} N = a - bM.\]


For Statisticians:

\[M_i - m_c | M_i > m_c \sim \text{Exp}(\beta).\]

Image source - Wikimedia

Biased estimation if rounding ignored.

All Together Now: ETAS model

\[ \lambda(t;\mathcal{H}_t, \theta) = \mu + \sum_{i:t_i < t} \kappa(m_i; K,a) h(t-t_i; c, p).\]

\[ \lambda(t;\mathcal{H}_t, \theta) = \mu + \sum_{i:t_i < t} K e^{a(m_i - m_c)} (p-1)c^{p-1} \left(1 + \frac{t-t_i}{c}\right)^{-p}.\]


\[ M_i \overset{\text{i.i.d}}{\sim} \text{Exp}(\beta). \]

ETAS Extensions

Image source - Wikimedia

ETAS Model Concerns

Veen and Schoenberg (2012) highlight several issues with the ETAS likelihood:


ETAS as a Branching Process

Conditional Inference

Conditioning on \(B\) provides very simple conditional distributions:

\[\begin{align*} \mu &| K, a, c, p, B - \text{Homogeneous PP}\\ K, a &| \mu, c, p, B - \text{Poisson Regression} \\ c, p &| \mu, K, a, B - \text{Power-law} \\ B &| \mu, K, a, c, p - B_i \text{ multinomial}. \end{align*}\]

Recent Developments

Unified Model and Dependence

Following from branching process representation Varty (2021) considers:

  • GPD as unified parametrisation.
  • Dependence between magnitudes.

Selecting M_c

Varty et al (2021) and Murphy et al (2023)

Errors in EV and PP models

There are an awful lot of measurement errors being ignored here… (Yue 2023-2025)

Future Directions

Wrapping Up

  1. Earthquakes are significant natural hazards but have received relatively little attention.

  2. State of the art models are reasonably accessible.

  3. Lots of contributions that could be made to extend these models by statistically minded folks.